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DISSIPATIVE BOUSSINESQ SYSTEM OF EQUATIONS

IN THE BENARD-MARANGONI PHENOMENON

R. A. Kraenkel, S. M. Kurcbart, J. G. Pereira

Instituto de Fısica Teorica

Universidade Estadual Paulista

Rua Pamplona 145,

01405-900 Sao Paulo SP – Brazil

M. A. Manna

Laboratoire de Physique Mathematique

Universite de Montpellier II

34095 Montpellier Cedex 05 – France

(February 9, 2008)

Abstract

By using the long-wave approximation, a system of coupled evolution

equations for the bulk velocity and the surface perturbations of a Benard-

Marangoni system is obtained. It includes nonlinearity, dispersion and dissi-

pation, and it can be interpreted as a dissipative generalization of the usual

Boussinesq system of equations. As a particular case, a strictly dissipative

version of the Boussinesq system is obtained. Finally, some speculations are

made on the nature of the physical phenomena described by this system of

equations.

PACS: 47.20.Bp ; 47.35.+i

Typeset using REVTEX

1

I. INTRODUCTION

Several recent works [1–8] have dealt with the study of oscillatory instabilities in sys-

tems of the Rayleigh-Benard and Benard-Marangoni type. Considering a shallow fluid layer

bounded below by a plane stress-free perfect conducting plate, and with a deformable upper

free surface where a constant heat flux is imposed, the linear stability analysis indicates that

oscillatory motion sets in when a certain critical combination of the Rayleigh and Marangoni

numbers is attained [1]. Subsequently, it was shown that, when the surface-tension can be

disregarded and the Rayleigh number of the system is at R = 30, appropriate surface deflec-

tions governed by the Korteweg-de Vries (KdV) equation appear [2]. For a two-dimensional

surface, the same deflections turn out to be governed by the Kadomtsev-Petviashvili (KP)

equation [3]. Thereafter, these results have been extended to the Benard-Marangoni sys-

tem [4,5], where buoyancy is discarded. In this case, appropriate surface disturbances prop-

agate according to the KdV equation when the Marangoni number of the system is at

M = −12. Further extentions to a double-diffusive systems [6], and to a temperature

depending viscosity [7] have also been established. The physical mechanism behind such

sustained excitations is the balance, at the critical point, between the energy dissipated by

viscous forces and that released either by buoyancy, or by a temperature depending surface

tension. In both cases, out of the critical points, appropriate surface excitations are governed

by the Burgers equation [5,8].

The KdV equation is well known to govern long surface-waves in inviscid shallow flu-

ids [9]. It corresponds to a situation where nonlinearity and dispersion compensate each

other, making possible the existence of coherent wave-structures, like the solitary-wave.

The reductive perturbation method of Taniuti [10], based on the concept of stretching, and

in which waves propagating in only one direction are sought from the beginning, is a com-

mon approach to study long waves in shallow water. From the various possible stretchings

of the coordinates, different evolution equations (linear wave, breaking wave, KdV) may

emerge as governing surface disturbances. However, there is an alternative approach to the

2

theory of long waves in shallow water, which is based on perturbative expansions in two

small parameters [9]. One of them measures the smallness of the amplitude perturbation,

and the other is a measure of the longness of the wave-length perturbation. This approach,

as an intermediate step, and from different possible relations between the two perturba-

tive parameters, yields different systems of evolution equations (nonlinear shallow water,

Boussinesq) describing superimposed waves propagating in opposite directions. Only when

specializing to waves moving in a given direction, equations like breaking wave and KdV

show up. A perturbative scheme of this kind has not been used to study surface excitations

in Benard systems. Consequently, for these systems, the more general evolution equations,

wherefrom Burgers and KdV equations are obtained, have not been found.

In this paper, instead of using the reductive perturbation method of Taniuti [10], we will

proceed through a perturbation scheme for the Benard-Marangoni system based on two per-

turbative parameters, leading to the so called long-waves in shallow water approximation.

By this way, a new system of coupled evolution equations will be found, involving the fluid

velocity and the free-surface displacement. This system may be interpreted as a dissipative

generalization of the Boussinesq equations. When the Marangoni number assumes the criti-

cal value M = −12, and a certain relation between the perturbative parameters is assumed,

it reduces, as we are going to see, to the usual Boussinesq equations. A further restriction

to waves moving in only one direction leads to the KdV equation. On the other hand, out

of the critical point, and assuming a different relation between the perturbative parameters,

the dissipative generalization of the Boussinesq equations reduce to a strictly dissipative

version of the Boussinesq equations. In this case, a restriction to waves moving in only one

direction will lead to the Burgers equation. Although we have restricted ourselves to the

Benard-Marangoni system, an extension to the Rayleigh-Benard case, where buoyancy is

predominant, may also be established with similar results [11]. Finally, through a linear

analysis, a speculation on the nature of the physical solutions to the strictly dissipative

Boussinesq system of equations is made.

3

II. BASIC EQUATIONS AND BOUNDARY CONDITIONS

We consider a fluid bounded below by a plane, stress-free, perfect termally conducting

plate at z = 0, and above by a deformable one-dimensional free surface which, at rest, lies at

z = d. The depth d will be supposed to be small enough so that buoyancy can be neglected

when compared to the effects coming from the surface tension dependence on temperature.

In other words, we will be dealing with a Benard-Marangoni system. The equations that

describe such a system are

~∇.~v = 0 , (1)

ρd~v

dt= −~∇p + µ∇2~v + ~gρ , (2)

dT

dt= κ∇2T , (3)

where ddt

= ∂∂t

+ ~v.~∇ is the convective derivative, ~v = (u, 0, w) is the fluid velocity, and p

is the pressure. The density ρ, the viscosity µ, and the thermal diffusivity coefficient κ are

supposed to be constant. The surface tension τ will be assumed to depend linearly on the

temperature:

τ = τ0[1 − γ(T − T0)] , (4)

where γ is a constant, and τ0, T0 are reference values for the surface tension and the tem-

perature, respectively.

The boundary conditions on the upper free surface z = d + η(x, t) are [12]

ηt + uηx = w , (5)

(p − pa) −2µ

N2

(

wz + uxη2

x − ηxuz − ηxwx

)

= −τ

N3ηxx , (6)

µ(

1 − η2

x

)

(uz + wx) + 2µηx (wz − ux) = N (τx + ηxτz) , (7)

4

ηxTx − Tz =F

kN , (8)

where F is the normal heat flux, k is the thermal conductivity, pa is a pressure exerted on

the upper surface, all of them supposed to be constant, and N = (1 + η2x)

1

2 . Subscripts

denote partial derivatives with respect to the corresponding coordinate.

On the lower plane z = 0, we assume that the sliding resistance between two portions

of the fluid is much greater than between the fluid and the plane [13], implying a stress-free

lower surface:

w = uz = 0 . (9)

Moreover, we will assume that the lower plate is at a constant temperature T = Tb.

III. PERTURBATIVE SOLUTION: EVOLUTION EQUATIONS

The static solution to the above equations is given by

ps = pa − ρg(z − d) ,

Ts = T0 −F

k(z − d) .

We consider now perturbations from this quiescent state. The horizontal and vertical length

scales of these perturbations are supposed to be l and a, respectively. Then, we define two

small parameters

ǫ =a

dδ =

d

l, (10)

which will be used to order the expansions. Before proceeding further, however, it is con-

venient to write the equations, boundary conditions and static solutions in a dimensionless

form. This is done by taking the original variables (primed) to be

x′ = lx z′ = dz t′ =l

c0

t (11a)

η′ = aη u′ =ag

c0

u w′ =ag

c0δw , (11b)

5

where c20 = gd. Furthermore, four dimensionless parameters appear: the Pradtl number

σ = µ/ρκ; the Reinolds number R = c0dρ/µ; the Bond number B = ρgd2/τ0, and the

Marangoni number M = γFd2τ0/kκµ.

To obtain the nonlinear evolution of the surface perturbations in the shallow water theory,

we expand all variables in powers of z, keeping the terms that will contribute to the evolution

equations up to orders ǫ and δ2. Despite laborious, this procedure is straightforward, and

for this reason we will omit the details, specifying in the text only the general guidelines,

and giving a brief summary of the results in the Appendix. To start with, we make the

expansions

u =∞∑

n=0

znun w =∞∑

n=0

znwn ,

where un and wn are both functions of x and t. Then, substituting them in Eq.(1), we get

the relation

wn+1 = −δ2unx

n + 1.

Using the boundary conditions at z = 0, it is easy to show that

w0 = w2 = w4 = ... = 0 ,

u1 = u3 = u5 = ... = 0 .

Then, using the expansion

p = ps +∞∑

n=0

znpn

in Eq. (2), it is possible to obtain u2, u4, u6 and p2 in terms of u0 and p0 only. The other

components of the expansions will contribute to orders higher than ǫ and δ2, and therefore

they can be neglected.

Next, expanding the temperature according to

T = Ts +∞∑

n=0

znθn ,

and using Eq. (3) with the corresponding boundary conditions, we can see that

θ0 = θ2 = θ4 = θ6 = ... = 0 .

6

Moreover, we can also obtain expressions for θ3 and θ5 in terms of u0 and p0 only. Now,

Eq.(6) yields p0 in terms of u0 and η. Consequently, it is possible to rewrite the u’s, p’s and

θ’s in terms of u0 and η only. Finally, using the above results in Eqs. (5) and (7), we obtain,

up to order ǫ and δ2, a coupled system of evolution equations for u0 and η:

u0t + ǫu0u0x + c2ηx −δ

R

(

4 +M

3

)

u0xx +δR

6(u0tt + ηxt) +

δ2R2

120(u0ttt + ηxtt)

− δ2

[

11

6−

M

30(4σ − 1)

]

u0xxt − δ2

(

1

B+

M

30+ 1

)

ηxxx = 0 , (12)

ηt + u0x + ǫ (u0η)x +δR

6(u0xt + ηxx) −

δ2

2u0xxx +

δ2R2

120(u0xtt + ηxxt) = 0 , (13)

where

c2 = 1 −M

σR2. (14)

The velocity u0 is only the first term in the expansion of u, which is :

u = u0 + δz2R2

2

(

u0t + ηx −3δ

Ru0xx

)

+ δ2z4R2

24(u0tt + ηxt) + O

(

ǫδ, δ3)

.

The value averaged over the depth is

u = u0 + δR

6(u0t + ηx) −

δ2

2

[

u0xx −R2

60(u0tt + ηxt)

]

+ O(

ǫδ, δ3)

.

The inverse is

u0 = u − δR

6(ηx + ut) +

δ2

2

[

uxx +7R2

180(ηxt + utt)

]

+ O(

ǫδ, δ3)

.

Substituting this in Eqs. (12) and (13), and using the lowest order equations into the uxxt

term, we obtain (omitting the tilde):

ut + c2ηx + ǫuux −δ

R

(

4 +M

3

)

uxx + δ2Ληxtt = 0 (15a)

ηt + ux + ǫ(uη)x = 0 , (15b)

where

Λ =4

3−

1

c2

(

1 +1

B

)

+M

30

(

1 −1

c2− 4σ

)

+1

6

(

4 +M

3

)(

1 −1

c2

)

. (16)

7

Due to the presence of the uxx term in Eqs. (15), this system can be considered as a

dissipative generalization of the Boussinesq equations. When the Marangoni number is at

the critical value M = −12, these equations coincide with the usual Boussinesq system of

equations

ut + c2ηx + ǫuux + δ2Ληxtt = 0 (17a)

ηt + ux + ǫ(uη)x = 0 . (17b)

In this case, by assuming that δ2 ≈ ǫ, and by specializing to waves moving, say, to the right,

we can obtain a relation between u and η:

u = cη − ǫc

4η2 − δ2

Λ

2ηxt , (18)

which is a kind of Riemann invariant [9]. Using this in Eq. (15) yields

ηt + cηx + ǫ3c

2ηηx + δ2

Λ

2ηxxx = 0 , (19)

where now

Λ =1

5

(

14

3+ 8σ

)

−1

c2

(

3

5+

1

B

)

. (20)

Equation(19) is the KdV equation.

Let us now consider the case M 6= −12. Assuming that δ ≈ ǫ, and neglecting terms of

order δ2 ≈ ǫ2, Eq. (15) becomes:

ut + c2ηx + ǫuux −δ

R

(

4 +M

3

)

uxx = 0 (21a)

ηt + ux + ǫ(uη)x = 0 . (21b)

This is a strictly dissipative version of the Boussinesq system, since the dispersive term is

not present now. By specializing again to waves moving to the right, we obtain the following

relation between u and η:

u = cη −ǫc

4η2 −

δ

2R

(

4 +M

3

)

ηx . (22)

8

Substituting this relation in Eq. (21), we obtain

ηt + cηx + ǫ3c

2ηηx −

δ

2R

(

4 +M

3

)

ηxx = 0 , (23)

which is the Burgers equation.

IV. THE STRICTLY DISSIPATIVE BOUSSINESQ EQUATION

The usual Boussinesq system of equations, Eqs. (17), may be transformed into the

classical Boussinesq equations, also known as dispersive long-wave equations [14]. These

equations have already been shown to be integrable [15]. The solutions to KdV and Burgers

equations have also been extensively discussed in the literature [16]. However, the strictly

dissipative Boussinesq system of equations, Eqs. (21), seems not to have been handled. The

existence of analytical solutions, therefore, is still an open issue.

The purpose of this section is to make some speculations on the nature of the physical

phenomena governed by the strictly dissipative Boussinesq system of equations. To start

with, we first rewrite Eqs. (21) in a dimensional form:

ut + c2gηx + uux − αdc0uxx = 0 (24a)

ηt + dux + uxη + uηx = 0 , (24b)

with

α =1

R

(

4 +M

3

)

,

and c2 given by Eq. (14). For M = −12, and changing the origin of the coordinates

according to

η(x, t) = h(x, t) − d ,

we obtain

ut + c2ghx + uux = 0 (25a)

ht + uxh + uhx = 0 (25b)

9

which, when c2 = 1, coincides with the classical one-dimensional shallow water system of

equations [9]. A general analytical solution to this system, which might present shocks, has

already been reported in the literature [17]. In fact, by restricting to waves moving to the

right we obtain the breaking-wave equation [9]

ht −c

2

√

gd hx +3c

2

√

g

dhhx = 0 . (26)

On the other hand, for M 6= −12, it is possible to gain some insight on the nature of

the solutions to Eqs. (24) by making a linear analysis. Neglecting the nonlinear term, Eqs.

(24) reads:

ut + c2gηx − αdc0uxx = 0 (27a)

ηt + dux = 0 . (27b)

Supposing solutions of the form exp(κx + ωt), we obtain the following dispersion relation:

ω = iαdc0κ

2

2±

αdc0κ2

2

[

−1 +4c2

α2d2κ2

]1/2

. (28)

There are three cases to be considered:

(i)When 4c2/α2d2κ2 < 1, then

κ >2c

αdor κ < −

2c

αd,

and the frequency ω is a pure imaginary number. In this case the system will present either,

a strictly dissipative or antidissipative behavior, depending on whether the imaginary part

of ω is negative or positive.

(ii)When 4c2/α2d2κ2 > 1, then

−2c

αd< κ <

2c

αd,

and the frequency ω is a complex number. In this case the system will present oscilla-

tory motion combined with either dissipation or anti-dissipation, depending on whether the

imaginary part of ω is negative or positive. For κ within this interval, the nonlinear dissi-

pative Boussinesq system, Eqs. (24), may possibly present a constant-profile opposite two

10

travelling-wave solutions if a compensation mechanism between dissipation and nonlinearity

takes place, in a way similar to what happens to the Taylor shock profile solution of Burgers

equation [9]. This, however, is still an open question.

(iii)When 4c2/α2d2κ2 = 1, then

κ = ±2c

αd,

and the frequency is again a pure imaginary number, ω = iωI , with

ωI =1

2αdc0κ

2 .

This is a transition case between the two cases previously described, in which the system

will again present either a strictly dissipative or anti-dissipative behavior, depending on

whether ωI is negative or positive. For these values of κ, as well as for the values of case (i),

the nonlinear dissipative Boussinesq system will present no oscillatory motion, which means

that its solutions must describe pure nonlinear diffusive phenomena at this region.

V. FINAL COMMENTS

The results obtained in this paper are two-fold. First, from a perturbative analysis,

corresponding to the long-wave in shallow-water approximation, we obtained a dissipative

generalization of the Boussinesq system of equations as governing the bulk velocity and

surface perturbations of a Benard-Marangoni phenomenon. This system of coupled evolution

equations includes nonlinearity, dispersion and dissipation. The predominance of any one of

them depends on the relation between ǫ and δ. Three cases are of special interest. First, when

M = −12, the dissipative term vanishes, and assuming that δ2 ≈ ǫ, the usual Boussinesq

system of equations is obtained. Second, when the Marangoni number is far enough from the

critical value, that is when M + 12 = O(1), and assuming now that δ ≈ ǫ, the dissipative

term dominates over the dispersive one. Then, by neglecting the dispersive term we got

Eq. (21), which is a strictly dissipative version of the Boussinesq system of equations. And

finally, as an intermediate case, we may also consider the situation in which M +12 = O(δ).

11

Assuming again that δ2 ≈ ǫ, it results that the dispersive and dissipative terms of Eq. (15)

are of the same order. In this case, no term is neglected, and the generalized Boussinesq

system, Eq. (15), will govern the bulk velocity and surface perturbations of the Benard-

Marangoni system. Upon specialization to waves moving, say, to the right, each one of these

three cases will lead to a single evolution equation for the surface displacement, which will

be, respectively, the KdV, Burgers and KdV-Burgers equations. It should be remarked that

the main feature of the perturbative scheme adopted here was to allow for a clear distinction

of the relative importance of each term in Eq.(15). By considering different relations between

the parameters δ and ǫ, a criteria to collect terms of the dominant order, and to neglect

higher order ones, was thus established.

The second result of this paper refers to the appearance of a new equation, which we have

called the strictly dissipative Boussinesq system of equations. It seems to be a nonintegrable

equation, and the existence of analytical solutions is still an open question. Despite of this

problem, we have performed here a linear analysis to get some insight on the nature of the

physical phenomena it might describe. Finally, due to the fact that it shows up in a physical

system, we believe this system of equation deserves further investigations as only few results

exist with respect to dissipative systems.

ACKNOWLEDGMENTS

The authors would like to thank Conselho Nacional de Desenvolvimento Cientıfico e

Tecnologico (CNPq), Brazil, for partial support. One of the authors (MAM) would like also

to thank the Instituto de Fısica Teorica, UNESP, for the kind hospitality, and the CNPq for

a travel grant.

APPENDIX:

We give here a summary of the results described, but not explicitly shown in the text.

The expressions for each term of the velocity expansion, already written in terms of u0 and

12

p0, and considering only those terms that will contribute to the evolution equations up to

orders ǫ and δ2, are

u2 =δR

2

(

u0t + ǫu0u0x +1

ǫp0x −

δ

Ru0xx

)

,

u4 =δ3R

24

(

−2

ǫp0xxx +

R

δu0tt +

R

ǫδp0xt − 2u0xxt

)

,

u6 =δ3R3

720

(

u0ttt +1

ǫp0xtt

)

.

The next terms of the expansion will not contribute to the evolution equations. In the same

way, the only term of the pressure expansion that will contribute is

p2 = −δ2

2p0xx .

From Eq. (3), with the corresponding boundary conditions, we can obtain expressions for

θ1, θ3 and θ5 in terms of u0 and p0 only:

θ1 = −ǫδRσ

2u0x + ǫδ2

R2σ

24(5σ − 1)u0xt − δ2

R2σ

24p0xx ,

θ3 = −ǫδ2R2σ2

12u0xt + ǫδ3

R2σ2

24(5σ − 1)u0xtt + ǫδ

Rσ

6u0x ,

θ5 = ǫδ2

[(

1 +1

σ

)

u0xt +1

ǫσp0xx − δ

Rσ

2u0xtt − δ

2

Rσu0xxx

]

.

Again, these are the only components of the temperature expansion that will contribute, up

to orders ǫ and δ2, to the evolution equations.

Now, from Eq. (6), we can get p0 in terms of u0 and η only:

p0 = ǫη − ǫδ2

Ru0x − ǫδ2

[

u0xt +(

1

B+

1

2

)

ηxx

]

+ ǫδ3

[

2

Ru0xxx −

R

12u0xtt

]

.

This equation allows us to rewrite the u’s, p’s and θ’s in terms of u0 and η. Consequently,

the system of evolution equations, Eq. (12), obtained from Eqs. (5) and (7) could also be

written in terms of u0 and η only.

13

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14

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15